The Four Color Map Problem

Suppose you have a map. Let's rule out degeneracies where a
country has separate parts (like the continental U.S. and
Alaska). Suppose you want to color all countries so they
are easy to distinguish. In particular you want to color
neighboring countries with different colors. How many
colors do you need at most? (Two countries are "
neighboring" if they share a border segment that
consists of more than one point. If sharing one point was
enough to be neighbors you could divide a pie into
arbitrarily many slices all of which share the center,
requiring as many colors as there are slices).

In 1852, Francis Guthrie wrote to his brother Frederick
saying it seemed that four colors were always sufficient,
did Frederick know a proof. Frederick asked his advisor
Augustus De Morgan. Morgan did not know either. In 1878
the mathematician Arthur Cayley presented the problem to the
London Mathematical Society. Less than a year later Alfred
Bray Kempe published a paper purporting to show that the
conjecture is true. His argument was considered correct
until 1890 when Percy John Heawood discovered a flaw. Work
by many people continued and the conjecture was finally
proved true in 1976 by Kenneth Appel and Wolfgang Haken.

The proof required unprecedented use of computer time and
takes up an entire book: Appel and Haken, Every Planar
Map is Four Colorable, Contemporary Mathematics, v.
98, American Mathematical Society, 1989, ISBN 0-8218-5103-9.
A very readable summary of the history and proof is in Appel
and Haken, The Solution of the Four-Color-Map Problem
, Scientific American, v. 237 (1977), No. 4, pp.
108-121.